In this contribution we address a-posteriori error estimation in $L^\infty(L^2)$ for a heterogeneous multiscale finite element approximation of time-dependent advection-diffusion problems with rapidly oscillating coefficient functions and with a large expected drift. Based on the error estimate, we derive an algorithm for an adaptive mesh refinement. The estimate and the algorithm are validated in numerical experiments, showing applicability and good results even for heterogeneous microstructures.